Abstract. We give a computational method for establishing a lower bound on the number of limit cycles bifurcating locally from a centre in a family of polynomial vector fields. When the sub-family of centers is known, this method can also be used to establish an upper bound to the number of limit cycles and prove rigorously the codimension of the family of centres.

As applications we give a new example of a cubic system with 11 local limit cycles and give lower bounds for the number of limit cycles in quartic systems.